Logic puzzles can teach reasoning in a fun way that doesn't feel like work.
- Logician Raymond Smullyan devised tons of logic puzzles, but one was declared by another philosopher to be the hardest of all time.
- The problem, also known as the Three Gods Problem, is solvable, even if it doesn't seem to be.
- It depends on using complex questions to assure that any answer given is useful.
Despite the general dislike of mathematics that most profess to have, many people enjoy logic puzzles. This is strange, as many logic puzzles are just variations of math problems. Gleefully ignorant of this fact, many mathaphobes will try to solve riddles and puzzles of tremendous difficulty using reasoning tools they fear to employ when the subject is an equation.
Today, we'll look at a puzzle, the polymath who devised it, and why you should consider picking up a book of logical puzzles next time you are at the library.
This puzzle was written by the brilliant logician Raymond Smullyan. Born in New York 101 years ago, Smullyan earned his undergraduate degree at the University of Chicago and his doctorate in mathematics at Princeton, where he also taught for a few years.
An extremely prolific writer, he published several books on logic puzzles for popular consumption and an endless stream of textbooks and essays for an academic audience on logic. His puzzle books are well regarded for introducing people to complex philosophical ideas, such as Gödel's incompleteness theorems, in a fun and non-technical way.
Skilled in close-up magic, Smullyan once worked as a professional magician. He was also an accomplished pianist and an amateur astronomer who built his own telescope. Besides his interest in logic, he also admired Taoist philosophy and published a book on it for a general audience.
He also found the time to appear on Johnny Carson, where, as in many of his books, he argued that people who like his puzzles claim to dislike math only because they don't realize that they are one and the same.
The Three Gods Problem
One of the more popular wordings of the problem, which MIT logic professor George Boolos said was the hardest ever, is:
"Three gods A, B, and C are called, in no particular order, True, False, and Random. True always speaks truly, False always speaks falsely, but whether Random speaks truly or falsely is a completely random matter. Your task is to determine the identities of A, B, and C by asking three yes-no questions; each question must be put to exactly one god. The gods understand English, but will answer all questions in their own language, in which the words for yes and no are da and ja, in some order. You do not know which word means which."
Boolos adds that you are allowed to ask a particular god more than one question and that Random switches between answering as if they are a truth-teller or a liar, not merely between answering "da" and "ja."
Give yourself a minute to ponder this; we'll look at a few answers below. Ready? Okay.
George Boolos' solution focuses on finding either True or False through complex questions.
In logic, there is a commonly used function often written as "iff," which means "if, and only if." It would be used to say something like "The sky is blue if and only if Des Moines is in Iowa." It is a powerful tool, as it gives a true statement only when both of its components are true or both are false. If one is true and the other is false, you have a false statement.
So, if you make a statement such as "the moon is made of Gorgonzola if, and only if, Rome is in Russia," then you have made a true statement, as both parts of it are false. The statement "The moon has no air if, and only if, Rome is in Italy," is also true, as both parts of it are true. However, "The moon is made of Gorgonzola if, and only if, Albany is the capitol of New York," is false, because one of the parts of that statement is true, and the other part is not (The fact that these items don't rely on each other is immaterial for now).
In this puzzle, iff can be used here to control for the unknown value of "da" and "ja." As the answers we get can be compared with what we know they would be if the parts of our question are all true, all false, or if they differ.
Boolos would have us begin by asking god A, "Does "da" mean yes if and only if you are True if and only if B is Random?" No matter what A says, the answer you get is extremely useful. As he explains:
"If A is True or False and you get the answer da, then as we have seen, B is Random, and therefore C is either True or False; but if A is True or False and you get the answer ja, then B is not Random, therefore B is either True or False… if A is Random and you get the answer da, C is not Random (neither is B, but that's irrelevant), and therefore C is either True or False; and if A is Random...and you get the answer ja, B is not random (neither is C, irrelevantly), and therefore B is either True or False."
No matter which god A is, an answer of "da" assures that C isn't Random, and a response of "ja" means the same for B.
From here, it is a simple matter of asking whichever one you know isn't Random questions to determine if they are telling the truth, and then one on who the last god is. Boolos suggests starting with "Does da mean yes if, and only if, Rome is in Italy?" Since one part of this is accurate, we know that True will say "da," and False will say "ja," if faced with this question.
After that, you can ask the same god something like, "Does da mean yes if, and only if, A is Random?" and know exactly who is who by how they answer and the process of elimination.
If you're confused about how this works, try going over it again slowly. Remember that the essential parts are knowing what the answer will be if two positives or two negatives always come out as a positive and that two of the gods can be relied on to act consistently.
Smullyan wrote several books with other logic puzzles in them. If you liked this one and would like to learn more about the philosophical issues they investigate, or perhaps if you'd like to try a few that are a little easier to solve, you should consider reading them. A few of his puzzles can be found with explanations in this interactive.
Do you ever act irrationally? You probably have. Let's take a look at how to fix that.
Most of us like to suppose that we are rational people, going about our days with at least some attempt at using logic and reason. However, logical fallacies and simple mistakes are everywhere. Some wrong ways of thinking are so familiar or so easy to overlook that it is possible you're unaware that there is even a mistake being made.
Here are ten logical fallacies and mistakes you make every day that cause life to be a little more difficult, and how you can avoid making those mistakes again.
The Gambler's Fallacy
When you flip a coin nine times in a row, can you use the results to predict what will happen the tenth time? While many people might try to say "tails has been on a streak" or "heads is overdue," neither of these past events has any effect on the next outcome. Both outcomes still have a 50-50 chance of happening on the next flip. The results of the next coin toss cannot be affected by the results of the last.
What should I do?
Instead of viewing probabilities in the long run, such as the idea that the coin has to have 50 heads and 50 tails results in a set of 100, or that a roulette wheel must hit all numbers at the same rate over a long enough time, look at each bet as separate from all others. The odds never change as a result of the last outcome for a fixed odds, random system.
The Appeal to Authority
Authority figures, but only on law.
Can something be true just because I say it is? Of course not. If your mechanic tells you that you need an oil change, is that true? It probably is. The appeal to authority is one of the subtler fallacies, but one that can still be overcome. Nothing is true just because an authority figure says it is. Instead, something is correct, and the authority figure has determined that fact by using their expertise on the subject.
Determining if the person you are talking to is trying to use raw, irrelevant authority to persuade you or if they actually are an expert on the subject is essential. The difficulty in saying that an authority figure is wrong was studied in the Milgram Experiment. However, it is rarely considered a good excuse to say you were just doing what you were told.
What should I do?
Don't blindly take a statement as true just because an authority figure gave it. My doctor is an authority on medicine and what he tells me about my health is likely to be correct. However, he has less knowledge when it comes to woodworking. On that subject, his authority as a doctor is meaningless. Always assure that an authority figure is qualified and that what they say is likely to be true before taking it as a fact.
The False Dilemma
We've all either heard or made this argument. We must do either A or B, and since A is not what we want then we must do B. However, very often we are facing a false dilemma. A situation where we have more than two choices and are being railroaded into thinking we don't.
What should I do?
When it seems you only have two choices, always make sure there are actually only two options. If a person starts a sentence with the phrase, "The choice is simple," know they are probably about to introduce a false dilemma.
The Post-Hoc Fallacy
Good luck charms, the most common form of this fallacy.
Many people tend to see patterns where they don't exist. This fallacy is when you connect two unrelated events and presume one caused the other. For example, when you flip on a light switch and hear a crash in the next room. Did flipping the switch cause the noise? No, but we often still try to connect events with no relationship. This fallacy is often the basis for good luck charms. "I brought my rabbit's foot with me, and it went well!" you might hear. But, it does not follow that the rabbit's foot caused the outcome.
What should I do?
Remember that coincidences sometimes happen and that sometimes two unrelated events can occur in a way to make them look related. Likewise, remember that one incident seeming to cause another wouldn't prove a relationship anyway; you would need many more tests to demonstrate that.
Affirming the Consequent
The building has collapsed, but do you know why?
This mistake is so easy to make that there can be no doubt that nearly everyone has done it. It is so similar to a valid form of thinking that the mistake can slip right past us.
While it is correct to argue this way:
If A, then B.
However, this is not correct:
If A, then B.
For example, saying "If the cornerstone is removed from the building it will fall over" is fine. But if we see the building has collapsed, it is still possible that another event caused it. The cornerstone might never have moved.
What should I do?
If-then thinking is beneficial and a useful tool, but always be sure that your thinking is going in the right direction. The cause can be used to predict the effect, but the result cannot be used to prove what the cause was. You need more evidence for that.
The Relativist Fallacy
If you believe it hard enough, is this dog really a unicorn?
Can the statement, "Well, it's true for me," ever be correct? It can, but you must use it carefully. While some statements are fully relative, like "I think cilantro tastes horrible," others are fully objective, like "Unicorns do not exist." While it makes sense for a person to say that cilantro tastes terrible to them, it doesn't work to say that unicorns are real for one person and not the next. The existence or non-existence of unicorns is an objective fact not influenced by any belief in that fact.
What should I do?
While some truths, such as ideas on what tastes good, are relative, others, such as what the capital of Canada is, are not. Before you either argue or listen to an argument that somebody is entitled to their own truth, first ask if the fact in question is one that can be relative. If that fact cannot be made true just by believing in it, then they this fallacy may be present.
The Genetic Fallacy
If I am made up of DNA, am I a double helix?
If one thing comes from another, do they have to share traits? This might seem like a convenient bias to have. However, do redwood trees seem to have much in common with their seeds? The genetic fallacy is the assumption that anything with an origin in one thing is highly likely to share traits.
What should I do?
This one is easy to do by accident, but also simple to overcome with a little extra thinking. Remember that things need not have the same traits as their origin. Think of the Volkswagen company; it was founded by the Nazi labor front. Does that make it a Nazi company now? Of course not, we would have to examine its present merits by themselves to determine that. The best thing to do for this fallacy is to try to examine why a thing has the traits it has without using its origin as an end-all answer.
The Inductive Fallacy
Will the sun always come up? It always has!
The sun came up today, does that mean it will come up tomorrow? David Hume showed us in 1748 that inductive arguments can never give us certainty, only probabilities and useful generalizations. The fact that apples always have fallen to the earth doesn't mean it will forever continue to happen. It is simply probable. Here's another example: "Harold is a grandfather. Harold is bald. Therefore, all grandfathers are bald." Inductive thinking makes a broad and highly probably generalization from specific information, but it is assumption, not certainty.
What should I do?
While you don't need to worry about the sun taking a day off tomorrow, it's not because it has never failed to rise. Inductive reasoning can't prove things, but it can be used to help find the best explanation for things. These reasons are better to use in arguments as to why an event will or will not happen than just saying that it has always happened before.
The Slippery Slope
A very slippery slope.
This fallacy is a common one. You have undoubtedly heard somebody say that taking action A is a slippery slope to taking action B, and B is horrible. They argue that we shouldn't take action A because it will, inevitably, lead us to take action B. But is that true? Generally speaking, no.
Now, slippery slope arguments can be good ones if it can be proved that the slope exists. If you can show that taking action A will inevitably lead to me taking action B then you have a good argument. However, most of the time people fail to demonstrate that inevitability.
What should I do?
If you are making the argument, be sure to demonstrate that action A concretely leads to action B. Simply saying "It could happen" doesn't count. You have to either prove it or show that it is much more likely to happen by action A taking place. If you are listening to the argument, always make sure that claimed connections between events are there.
The Masked Man Fallacy
A masked man.
Identical objects share all of the same properties. This rule, called Leibnitz's law, seems simple enough to understand. However, it is very easy to misuse this concept to make bad arguments.
This argument is correct:
1. A is C
2. B is not C
Therefore: A is not B.
However, you can't plug in just any property into the argument and have it work. Think about this one:
The Joker believes that Batman beat him up.
The Joker does not believe that Bruce Wayne beat him up.
Therefore: Batman is not Bruce Wayne.
While physical properties follow Leibnitz's law, attitudes, beliefs, and psychological states don't necessarily do so.
What should I do?
When you are identifying a person, object, or idea be sure to check that the properties you are looking for are non-relative ones.
Here are more tips for making better decisions, from poker pro Liv Boeree:
Author, speaker, and public intellectual Richard Dawkins is a first-class debater on subjects as grand and reaching as the very existence (or lack thereof) of a master creator. But he's got a simple yet highly effective technique to win people over to see his point of view. Find out what it is right here.
Many people would like to have a one-on-one argument with renowned professor, author, and all-around big thinker Richard Dawkins. He's most one of the world's most prominent public intellectuals and has written over a dozen books on matters as wide-ranging as atheism and science. Because he attacks such deeply held beliefs, many people disagree with him. But how is he so effective at what he does? Simple. He imagines his argument from the other side's perspective. That way, Richard Dawkins posits, there's a much higher chance that he can land his point. Richard Dawkins' new book is Science in the Soul: Selected Writings of a Passionate Rationalist.
Is creativity a wild and free state of mind, or is it actually a pattern that others just can't recognize?
To ensure your survival, your brain evolved to avoid one thing: uncertainty. As neuroscientist Beau Lotto points out, if your ancestors wondered for too long whether that noise was a predator or not, you wouldn't be here right now. Our brains are geared to make fast assumptions, and questioning them in many cases quite literally equates to death. No wonder we're so hardwired for confirmation bias. No wonder we'd rather stick to the status quo than risk the uncertainty of a better political model, a fairer financial system, or a healthier relationship pattern. But here's the catch: as our brains evolved toward certainty, we simultaneously evolved away from creativity—that's no coincidence; creativity starts with a question, with uncertainty, not with a cut and dried answer. To be creative, we have to unlearn millions of years of evolution. Creativity asks us to do that which is hardest: to question our assumptions, to doubt what we believe to be true. That is the only way to see differently. And if you think creativity is a chaotic and wild force, think again, says Beau Lotto. It just looks that way from the outside. The brain cannot make great leaps, it can only move linearly through mental possibilities. When a creative person forges a connection between two things that are, to your mind, so far apart, that's a case of high-level logic. They have moved through steps that are invisible to you, perhaps because they are more open-minded and well-practiced in questioning their assumptions. Creativity, it seems, is another (highly sophisticated) form of logic. Beau Lotto is the author of Deviate: The Science of Seeing Differently.
You really do have to know when to hold 'em, and know when to fold 'em, and most of the time trusting your gut is a copout, says poker champion Liv Boeree.
Want to win at poker? First, understand how your mind works. In Thinking, Fast and Slow, Nobel-Prize winning psychologist Daniel Kahneman explains the mind’s two thinking systems: System 1 which "is the brain’s fast, automatic, intuitive approach," and System 2 which is "the mind’s slower, analytical mode, where reason dominates." Champion poker player Liv Boeree explains that if you want to rake in the chips, you’ll need to harness both these systems. The game requires players to make decisions in circumstances of great uncertainty, so learn to balance your mathematical reasoning with your gut instincts—however, when those two systems give you strongly conflicting messages, Boeree says the win is usually based in logic, not intuition. Your gut is valuable, and can often be the key to reading a tell or bluff, but it "isn’t as reliable as you think." Find out more at www.livboeree.com.